# Definition:Calculus/Integral

## Definition

**Integral calculus** is a subfield of calculus which is concerned with the study of the rates at which quantities accumulate.

Equivalently, given the rate of change of a quantity **integral calculus** provides techniques of providing the quantity itself.

The equivalence of the two uses are demonstrated in the Fundamental Theorem of Calculus.

The technique is also frequently used for the purpose of calculating areas and volumes of curved geometric figures.

## Also see

- Results about
**integral calculus**can be found**here**.

## Historical Note

The origins of **integral calculus** can be traced back to the ancient Greek mathematicians' attempts to calculate the area of a circle.

Eudoxus of Cnidus may have been the earliest such, with his method of exhaustion, dating from about $360$ BCE.

The techniques were used and expanded upon in his work *The Method* by Archimedes of Syracuse, to calculate areas and volumes of curvilinear figures.

These techniques are often suggested as being the precursors to **integral calculus**.

Johannes Kepler, in his *Nova Stereometria Doliorum Vinariorum* of $1615$, devised a method of finding the volume of a solid of revolution by slicing it into thin disks, calculating the volume of each, and then adding those volumes together.

Bonaventura Francesco Cavalieri expanded upon this in his *Geometria Indivisibilibus Continuorum Nova Quadam Ratione Promota* of $1635$.

John Wallis arithmetized Cavalieri's ideas in his *Arithmetica Infinitorum* of $1656$.

Much of the early work developing **integral calculus** was done by Isaac Newton.

His initial work on this seems to have been achieved during the years $1665$ to $1667$ when he was at home in Woolsthorpe.

At the same time that Newton was arranging his thesis, Gottfried Wilhelm von Leibniz was publishing many papers himself on the same subject.

The rigorous treatment of the subject was developed later, by Carl Friedrich Gauss, Niels Henrik Abelâ€Ž and Augustin Louis Cauchy.

## Linguistic Note

The term **integral calculus** is the Anglified version of the neo-Latin phrase **calculus integralis**, made popular by Gottfried Wilhelm von Leibniz.

At first he suggested **calculus summatorius**, but in $1696$ he decided with Johann Bernoulli that **calculus integralis** would be better.

Some sources suggest that the name **integral**, first used by him in $1690$, may well have been coined by Jacob Bernoulli.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies - 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{VI}$: On the Seashore - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**integral calculus** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**calculus** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**integral calculus** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**calculus** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**integral calculus** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $8$: The System of the World: Calculus - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**integral calculus**